Friday, September 14, 2012

Mathematicians care about proofs

In my FQXi essay, I argue that mathematics is the same as what is provable in Zermelo–Fraenkel set theory (ZFC), and hence subject to the limitations of such proofs. This view is conventional wisdom, so I did not give much support for it. But I just saw another view on another blog.

The (really really) big news in the math world today is that Shin Mochizuki has (plausibly) claimed to have solved the ABC problem, which in turn suffices to settle many of the most vexing outstanding problems in arithmetic. ...

No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth. ...

Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.

That in turn implies that there is such a thing as mathematical truth, independent of what we can prove.

No, this is contrary to two millennia of mathematical wisdom. Mathematicians only care about what is provable. For them, the truth is what is provable. Nobody changes the rules of the game.

Since Euclid axiomatized geometry, a geometry truth meant a theorem with a proof from the axioms. Sure, people have invented other geometries with other axioms, but that has not changed fundamental ideas about mathematical truth or the necessity of proof.

In the following discussion, we shall work with various models — consisting of “sets” and a relation “∈” — of the standard ZFC axioms of axiomatic set theory [i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice — cf., e.g., [Drk], Chapter 1, §3]. We shall refer to such models as ZFC-models. ...

We shall refer to a ZFC-model that also satisfies this additional axiom of the Grothendieck school as a ZFCG-model. ...

Although we shall not discuss in detail here the quite difficult issue of whether or not there actually exist ZFCG-models, we remark in passing that one may justify the stance of ignoring such issues — at least from the point of view of establishing the validity of various “final results” that may be formulated in ZFC-models — by invoking a result of Feferman [cf. [Ffmn], §2.3] concerning the “conservative exten- sionality” of ZFCG relative to ZFC, i.e., roughly speaking, that “any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model in fact holds in the original ZFC-model”.

So in fact Mochizuki cares very much about proving his theorems in ZFC, even if he likes to use tools that go outside ZFC. Mathematical truth is what is provable in ZFC.

Update: In the comments on his blog, Landsburg says that algebraic geometers cheerfully cite the work of Grothendieck without worrying about foundational issues. Sure, mathematicians like to specialize, and most of them do not study foundations. Algebraic geometers especially have a cult-like reputation of ignoring others. The Italian school of algebraic geometry is famous for making sloppy assumptions that are sometimeswrong. But even still, today's algebraic geometers do not condone changing the rules of the game because some (alleged) truth is not provable.

Saying that algebraic geometers cite Grothendieck without worrying about foundations is about like saying that they don't worry about differential equations. Yeah, they would rather let someone else worry about the differential equations. But that does not mean that they don't care about what is provable.

Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and Deligne (1977) gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved in ZFC) led to some uninformed speculation that étale cohomology and its applications (such as the proof of Fermat's last theorem) needed axioms beyond ZFC. In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC (and even in much weaker theories).

Ulrik Buchholtz on September 7, 2012 said:
It seems to me that Mochizuki defines species, mutation and morphism-of-mutation as being whatever defines a category, functor or natural transformation in any model of ZFC. Thus, by the completeness theorem, these are nothing but formalizable-in-ZFC categories, functors, and natural transformations. ...

On another note, on p. 43 Mochizuki seems to claim that Grothendieck set theory is a conservative extension of ZFC, which it certainly is not. What the Feferman-paper he refers to shows, is that an extension of ZFC with a constant for a universe satisfying schematic reflection, ZFC/s, is a conservative extension of ZFC, and that’s obviously because every model of ZFC can be extended to a model of ZFC/s (using a Löwenheim-Skolem construction), but a model of ZFC/s need not be a model of Grothendieck set theory.

Terence Tao on September 8, 2012 said:
On reading the Mochizuki paper, it seems to me (if I am not mistaken) that all the set theoretic and model theoretic stuff is really only used in Section 3, whereas the ABC conjecture is proven (at least allegedly) by Section 2. As far as I can tell, there are no forward references to Section 3 in previous sections other than side remarks. So perhaps all the set and model theory here is in fact something of a red herring as far as the application to ABC is concerned, and are primarily relevant for further development of Mochizuki’s inter-universal geometry instead? (Among other things, this would render the issue of the non-conservative nature of Grothendieck set theory somewhat moot.)

The point remains that all these mathematicians are interested in proving the abc conjecture in ZFC. If mathematicians were really willing to lower their standards of proof, then they would accept this heuristic argument. More info on the proposed proof is here.

Yes, we accept expert opinion that FLT has been proved. But if the experts said, "we could not prove FLT but we decided that it was true so we changed the rules of the game", then we would not accept FLT.

"No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth."

There are parts of this statement that I accept and other parts that I reject. I agree that mathematicians are interested in truth, but I don't think that mathematics has an "objective" or "absolute" notion of truth. In mathematics, truth is always relative to the assumptions you start with. If you want to do mathematics in ZFC, then that's great. There's lots of wonderful wonderful mathematics in ZFC. If instead you want to do mathematics in ZF and assume the negation of the axiom of choice, that's another perfectly interesting and worthwhile way of doing mathematics. Similarly, you can assume the continuum hypothesis is true, or you can assume it's false. There is no "absolute" sense in which a statement like the continuum hypothesis is true or false.

"Nobody changes the rules of the game."

This statement is completely false and does not reflect what mathematicians actually do. Mathematicians have studied alternative foundations for mathematics based on category theory and Grothendieck universes, for example. Modern set theorists experiment with large cardinals all the time. The fact that Mochizuki is working with alternative foundations for mathematics is further evidence that mathematicians can and do change the rules of the game.

"So in fact Mochizuki cares very much about proving his theorems in ZFC, even if he likes to use tools that go outside ZFC."

In some parts of algebraic geometry, you want to formalize certain intuitions about space, and going outside of ZFC turns out to be quite convenient. Going outside of ZFC can also produce many rich and interesting results with applications. So what's the point of restricting ourselves to ZFC? I don't see anything wrong with exploring alternative foundations as long as we remember which foundation of mathematics we're using.

The point of restricting to ZFC is to get a publishable proof that is acceptable to mathematicians. Sure you can assume the Continuum Hypothesis or the Riemann Hypothesis, but then your theorems will only be publishable of the form "if CH then XYZ." If it turns out that Mochizuki has to assume axioms outside ZFC, then he will not be considered to have proved the abc conjecture.

If I am wrong, then give me an example of some theorem outside ZFC that mathematicians accept. Sure category theory and Grothendieck universes are formally outside ZFC, but they are being used to prove things that can be proved in ZFC. Logicians experiment with large cardinals, but they do not change the rules of the game. They might say, "It is a theorem of ZFC that if there is an inaccessible cardinal them ..." Give me some example of mathematicians changing the rules of the game to expand what is true or provable.

"but then your theorems will only be publishable of the form 'if CH then XYZ.'"

The point is that all mathematical results are of this form. When you publish a result, you're always explaining the consequences of some set of axioms, either ZFC or something else. Since mathematical results are contingent on the assumptions they make, they are always of the form "if A then B". They never assert the truth of a proposition in some absolute, objective sense, independently of any axioms.

"If it turns out that Mochizuki has to assume axioms outside ZFC, then he will not be considered to have proved the abc conjecture."

If you define the abc conjecture to be a statement about the consequences of the ZFC axioms, then no, he would not have proved the abc conjecture. But even if he's using new foundations which are not equivalent to ZFC, his work still shows that this alternative foundation has interesting and highly nontrivial consequences. That's why mathematicians are taking him seriously.

"If I am wrong, then give me an example of some theorem outside ZFC that mathematicians accept."

If you're asking for a theorem outside of ZFC which is true in the sense that it follows from the axioms of ZFC, then obviously that's an impossible task. But I've already given you many examples of topics which are considered legitimate parts of mathematics even though they are not based on ZFC...

The question here is whether a non-ZFC argument would ever cause the abc conjecture to be accepted as being true. I say no, and I say that nothing like that has ever happened in the history of mathematics.

The statements of the Millennium Prize Problems say nothing about the proofs being in ZFC, but it is understood. A non-ZFC proof would not be accepted.

If I am wrong, then give me some example. The closest example I know is Wiles's proof of Fermat's Last Theorem, which was accepted in spite of its apparent dependence on Grothendieck universes. But the experts were satisfied that the proof could be formalized in ZFC. No one said that we need to accept a non-ZFC proof because truth required changing the rules of the game.

I don't understand what you're asking for an example of. Any result on large cardinals would be accepted as "true" by mathematicians. Such a result wouldn't be "ZFC-true" but it would still be true in the sense that it was a correct consequence of some clearly stated axioms.

It seems like you want an example of a question about ZFC which was answered using non-ZFC assumptions. Obviously such an example doesn't exist. If we interpret the abc conjecture as a question about the validity of a certain statement in the ZFC framework, then a proof cannot use axioms that are outside of ZFC. The point I've been making is that there's much more to mathematics than just ZFC.

The abc conjecture is a statement about the integers. What would convince mathematicians that it is true? I say only a ZFC proof. It would be interesting if someone proved abc as a consequence of some large cardinal axiom, but that would not be accepted as a proof of abc unless it could be formalized in ZFC.

If I am wrong here, then there should be some example of a non-ZFC proof being accepted. I do not know of any such example.